Great Expectations. Part I: On the Customizability of Generalized Expected Utility

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Great Expectations.

Part I: On the Customizability of Generalized Expected Utility

∗

Francis C. Chu Joseph Y. Halpern

Department of Computer Science Cornell University

Ithaca, NY 14853, U.S.A. Email: {fcc,halpern}@cs.cornell.edu

November 5, 2004

Abstract

We propose a generalization of expected utility that we call generalized EU (GEU), where a decision maker’s beliefs are represented by plausibility measures and the decision maker’s tastes are represented by general (i.e., not necessarily real-valued) utility functions. We show that every agent, “rational” or not, can be modeled as a GEU maximizer. We then show that we can customize GEU by selectively imposing just the constraints we want. In particular, we show how each of Savage’s postulates corresponds to constraints on GEU.

1 Introduction

Many decision rules have been proposed in the literature. Perhaps the best-known approach is based on maximizing expected utility (EU), calculated either with respect to a given (objective) probability measure, as done originally by von Neumann and Morgenstern [1947], or with respect to a probability measure constructed from a preference relation on alternatives that satisfies certain postulates, as done originally by Savage [1954]. All these approaches follow the same pattern: they formalize the set of alternatives among which the decision maker (DM) must choose (typically as acts or lotteries). They then give a set of assumptions (often called postulates or axioms) such that the DM’s preferences on the alternatives satisfy these assumptions iff the preferences have an EU representation, where an EU representation of a preference relation is basically a utility function (and a probability measure when acts are involved) such that the relation among the alternatives based on expected utility agrees with the preference relation. Moreover, they show that the representation is essentially unique, in the sense that, given two representations of the same preference relation, the utility functions are positive affine transformations of one another and the probability measures are equal. Thus, if the preferences of a DM satisfy the assumptions, then she is behaving as if she has quantified her tastes via a real-valued utility function (and her beliefs via a probability measure) and she is relating the alternatives according to their expected

∗

Work supported in part by NSF under grants IIS-0090145 and CTC-0208535 and by the DoD Multidisciplinary University Research Initiative (MURI) program administered by ONR under grant N00014-01-1-0795. A preliminary version of this paper appeared in Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI 2003), 2003.

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utility. These assumptions are typically regarded as criteria for rational behavior, so these results also suggest that if a DM’s beliefs are actually described by a probability measure and her tastes are described by a utility function, then she should relate the alternatives according to their expected utility (if she wishes to appear rational).

Despite the appeal of EU maximization, it is well known that people do not follow its tenets in general [Resnik 1987]. As a result, a host of extensions of EU have been proposed that accommodate some of the more systematic violations (see, for example, [Gul 1991; Gilboa and Schmeidler 1989; Giang and Shenoy 2001; Kahneman and Tversky 1979; Luce 2000; Quiggin 1993; Schmeidler 1989; Tversky and Kahneman 1992; Yaari 1987]). For each of these approaches, a representation theorem is proved. These representation theorems essentially view a decision rule R as a function that maps tastes (and perhaps beliefs, depending on the rule) to a preference relation on alternatives. The theorems then say that, for some fixed rule R (say EU), there is a set AR of assumptions about

preference relations (such as the von Neumann-Morgenstern axioms or the Savage postulates) such that a preference relation - satisfies AR iff there exist some tastes (and beliefs) such that, given

these as inputs, R returns -. In other words, if the preference relation of an agent satisfies AR,

then she behaves as if she is using R, and a user of R will always exhibit preference relations that satisfies AR.

Given this plethora of rules, it would be useful to have a general framework in which to study decision making. The framework should also help us understand the relationship between various decision rules. We provide such a framework in this paper.

The basic idea of our approach is to generalize the notion of expected utility so that it ap-plies in as general a context as possible. To this end, we allow DMs to express their beliefs using plausibility measures [Friedman and Halpern 1995; Halpern 2003], a generalization of probability measures, Choquet capacities, and many other representations of uncertainty. A plausibility mea-sure associates with each event its plausibility, an element of an arbitrary partially ordered space. Similarly, DMs express their tastes using elements of an arbitrary partially-ordered utility domain. (Indeed, our results apply even if the relation on utilities is not transitive.) We then assume that there are functions ⊕ and ⊗ that serve as analogues of + and ×; ⊗ combines a plausibility and a utility to give a value, ⊕ combines two values to give another value. Finally, we assume a binary relation - over values. In this setting, we can define a generalization of expected utility, which we call generalized EU (GEU). The GEU of an act is basically the sum (in the sense of ⊕) of products (in the sense of ⊗) of plausibility values and utility values that generalizes the standard definition of (probabilistic) expected utility over the reals in the obvious way.

We start by proving an analogue of Savage’s result with respect to the decision rule (Maximizing) GEU. (In this paper we use “Maximizing GEU” and “GEU” interchangeably.) We show that every preference relation on acts has a GEU representation (even those that do not satisfy any of Savage’s postulates), where a GEU representation of a preference relation basically consists of a choice of a plausibility measure, a utility function, functions ⊕ and ⊗, and a binary relation - on values (Theorem 3.1). In other words, no matter what the DM’s preference relation on acts is, she behaves as if she has quantified her beliefs by a plausibility measure and her tastes via a utility function, and is relating the acts according to their (generalized) expected utility as defined by ⊕ and ⊗. That is, we can model any DM using GEU, whether or not the DM satisfies any rationality assumptions. Given that GEU can represent all preference relations, it might be argued that GEU is too general—it offers no guidelines as to how to make decisions. We view this as a feature, not a bug, since our goal is to provide a general framework in which to express and study decision rules, instead of proposing yet another decision rule. Thus the absence of “guidelines” is in fact an absence of limitations: we do not want to exclude any possibilities at the outset, even preference relations that are not transitive or are incomplete. From the point of view of a behavioral scientist, this has the

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advantage of allowing us to represent the preference relations that actually arise in real life, which typically do not satisfy many of the standard assumptions made by decision theorists, and doing so in a potentially compact way (by specifying ⊕, ⊗, -, a plausibility measure Pl, and a utility function u). More interestingly, starting from a framework in which we can represent all preference relations, we can then consider what preference relations have “special” representations, in the sense that the expectation domain, plausibility measure, and utility function in the representation satisfy some (joint) properties. This allows us to show how properties of ⊕, ⊗, and - correspond to properties of preference relations. We can then “customize” GEU by placing just the constraints we want. We illustrate this by showing how each of Savage’s postulates corresponds in a precise sense to an axiom on GEU.

This ability to customize GEU may be of particular interest in the design of software agents that make decision on our behalf. It may not be appropriate to assume that such software agents represent beliefs using probability measures and tastes using real-valued utilities. For example, the information that a system can obtain may be better modeled by a set of probability measures than a single probability measure, and a user may represent his or her tastes more qualitatively, using words like “terrific” and “terrible”, rather than numerically. Using GEU, it should be possible to design agents that make decisions based on more general representations of beliefs and tastes, and customize them so that the the decision-making process satisfies certain “rationality” postulates.

There is yet another advantage of this approach, which is the focus of [Chu and Halpern 2003]. Intuitively, a decision rule maps tastes (and beliefs) to preference relations on acts. Given two decision rules R1and R2, an R1 representation of R2is basically a function τ that maps inputs of R2

to inputs of R1that represent the same tastes and beliefs, with the property that R1(τ (x)) = R2(x).

Thus, τ models, in a precise sense, a user of R2as a user of R1, since τ preserves tastes (and beliefs).

In [Chu and Halpern 2003] we show that many decision rules have GEU representations. Moreover, we show that (almost) every decision rule has an ordinal GEU representation, where, rather than x and τ (x) representing exactly the same tastes (and beliefs), they preserve the relation between tastes (and beliefs), without necessarily preserving the magnitude. For example, if outcome o1 is

preferred to outcome o2in x, o1 is also preferred to o2in τ (x), although the magnitude of preference

may be different.1 The bottom line here is that essentially all rules can be viewed as special cases of GEU.

The rest of this paper is organized as follows. We cover some basic definitions in Section 2: plausibility domains, utility domains, expectation domains, decision problems, and GEU. We show that every preference relation on acts has a GEU representation in Section 3. In Section 4, we show that each of Savage’s postulates corresponds to an axiom on GEU. We conclude in Section 5. Most proofs are deferred to the appendix.

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Our claim that only “many” decision rules have a GEU representation may seem inconsistent with our earlier claim that every preference relation has a GEU representation. However, there is no inconsistency, since representing a decision rule is not the same as representing a preference relation. If we again view a decision rule as a function from tastes (and possibly beliefs) to preference relation on alternatives, then a decision rule R represents a preference relation - if there are some tastes and beliefs such that, with these as input, R returns -. On the other hand, R1 represents R2 if, roughly speaking, for all possible inputs of tastes (and beliefs), R1 and R2 return the same

preference relation. That is, R1and R2 act essentially the same way as functions on the domain of R2(R1 can have

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2 Preliminaries

2.1 Decision Situations, Expectation Domains, and Decision Problems

A decision situation (under uncertainty) describes the objective part of the circumstance that the DM faces (i.e., the part that is independent of the tastes and beliefs of the DM). We model a decision situation in a standard way, as a tuple A = (A, S, C), where

• S is the set of states of the world, • C is the set of consequences, and

• A is a set of acts (i.e., a set of functions from S to C).

An act a is simple iff its range is finite. That is, a is simple if it has only finitely many consequences. Many works in the literature focus on simple acts (e.g., [Fishburn 1987]). We assume in this paper that A contains only simple acts; this means that we can define (generalized) expectation using finite sums, so we do not have to introduce infinite series or integration for arbitrary expectation domains. Note that all acts are guaranteed to be simple if either S or C is finite, although we do not assume that here.

Since one of the goals of this paper is to provide a general framework for all of decision theory, we want to represent the tastes and beliefs of a DM in a decision situation in as general a framework as possible. In particular, we do not want to force the DMs to linearly preorder all consequences and all events. Thus, as discussed in the introduction, we use plausibility measures to represent the beliefs of the DMs and (generalized) utility functions to represent their tastes.

A plausibility domain is a set P , partially ordered by P (so P is a reflexive, antisymmetric,

and transitive relation), with two special elements ⊥P and >P, such that ⊥P P x P >P for all

x ∈ P . (We often omit the subscript P in ⊥P and >P when it is clear from context.) Given a set

S of states, a function Pl : 2S → P is a plausibility measure iff Pl1. Pl(∅) = ⊥,

Pl2. Pl(S) = >, and

Pl3. if X ⊆ Y then Pl(X) Pl(Y ).

Clearly plausibility measures are generalizations of probability measures. As pointed out in [Friedman and Halpern 1995], plausibility measures generalize a host of other representations of uncertainty as well. Note that while the probability of any two sets must be comparable (since R is totally ordered), the plausibility of two sets may be incomparable.

We also want to represent the tastes of DMs using something more general than R, so we allow the range of utility functions to be utility domains, where a utility domain is a set U endowed with a reflexive binary relation -U. Intuitively, elements of U represent the strength of likes and dislikes

of the DM while elements of P represent the strength of her beliefs. Note that we do not require the DM’s preference to be transitive (although we can certainly add this requirement). Experimental evidence shows that DM’s preferences occasionally do seem to violate transitivity.

Once we have plausibility and utility, we want to combine them to form expected utility. To do this, we introduce expectation domains, which have utility domains, plausibility domains, and operators ⊕ (the analogue of +) and ⊗ (the analogue of ×).2 More formally, an expectation domain is a tuple E = (U, P, V, ⊕, ⊗), where (U, -U) is a utility domain, (P, P) is a plausibility domain,

2

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(V, -V) is a valuation domain (where -V is a reflexive binary relation), ⊗ : P × U → V , and

⊕ : V × V → V . The following four requirements must hold for all x, y, z ∈ V and u ∈ U : E1. (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z);

E2. x ⊕ y = y ⊕ x; E3. > ⊗ u = u;

E4. (U, -U) is a substructure of (V, -V).

E1 and E2 say that ⊕ is associative and commutative. E3 says that > is the left-identity of ⊗ and E4 ensures that the expectation domain respects the relation on utility values.

Note that we do not require that ⊕ be monotonic; that is, we do not require that, for all x, y, z ∈ V ,

if x -V y then x ⊕ z -V y ⊕ z. (2.1)

We say that E is monotonic iff (2.1) holds. It turns out that monotonicity does not really make a difference by itself (see Corollary 3.2), but if we also assume that every element of the valuation do-main has a ⊕ inverse, then monotonicity would limit the preference relations that are representable by GEU.

Recall that in the standard case, ⊥ = 0 and 0 × x is the identity for +. In general, we do not assume that ⊥ ⊗ u is the identity ⊕ (or that ⊕ even has an identity). We say that E has a standard ⊕ identity iff

(⊥ ⊗ u) ⊕ x = x for all u ∈ U and x ∈ V . (2.2) Because ⊕ is commutative, there can clearly be at most one identity for ⊕, so if (2.2) holds, then ⊥ ⊗ u1= ⊥ ⊗ u2 for all u1, u2 ∈ U . (In many cases of interest, there is a natural identity for ⊕, and

it is the standard identity; e.g., see Examples 2.1 and 2.2.) Requiring (2.2) has very little effect on our results. Some of the proofs become easier, while others become somewhat more difficult, but the theorems still hold.

Note that we also do not require ⊗ to distribute over ⊕. The obvious way to state such a requirement is to require that p ⊗ (x ⊕ y) = (p ⊗ x) ⊕ (p ⊗ y). However, this is not well-defined in general, since the domain of ⊗ is P × U and the domain of ⊕ is V × V . If x, y ∈ U , then x ⊕ y, p ⊗ x, and p ⊗ y are all well defined, but x ⊕ y may be an element of V − U , so p ⊗ (x ⊕ y) may not be well defined. As we shall see, in cases where the distributive property makes sense (for example, if V = U or if u1⊕ u2 ∈ U for all u1, u2 ∈ U ), then it actually does hold in many examples of

interest.

Example 2.1 The standard expectation domain, which we denote E, is (R, [0, 1], R, +, ×), where the ordering on each domain is the standard order on the reals. This, of course, is the expectation domain which is used in defining most decision rules in the literature. It is clearly monotonic and has a (standard) + identity, namely 0.

Example 2.2 Consider the expectation domain E2 = (R, [0, 1] × [0, 1], R × R, ⊕, ⊗), where

• we use the standard order on the utility domain R;

• the order on the plausibility domain [0, 1] × [0, 1] is such that (p₁, p2) (q1, q2) iff p1≤ q1

and p2 ≤ q2;

• similarly, (u1, u2) -V (v1, v2) iff u1 ≤ v1 and u2≤ v2;

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• ⊗ is pointwise multiplication: (p₁, p2) ⊗ u = (p1× u, p2× u).

We can view the utility domain R as a substructure of the valuation domain R × R by identifying the element u ∈ R with the pair (u, u). Note that the ordering on the plausibility domain and the ordering on the valuation domain are both partial. E2 is also monotonic, and has a standard

⊕ identity—(0, 0). The distributive property (which makes sense here) is also easily seen to hold: (p1, p2) ⊗ (u1⊕ u2) = ((p1, p2) ⊗ u1) ⊕ ((p1, p2) ⊗ u2).

It turns out to also be of interest to consider the expectation domain E₂0, which is defined just like E2, except that the order -0V on the valuation domain is defined by taking (u1, u2) -0V (v1, v2)

iff min(u1, u2) ≤ min(v1, v2). Note that this makes -0_V a total preorder.

A decision problem is essentially a decision situation together with information about the tastes and beliefs of the DM; that is, a decision problem is a decision situation together with the subjective part of the circumstance that faces the DM. As we said earlier, we use (generalized) utility functions and plausibility measures to represent these tastes and beliefs; thus, we must specifically include an expectation domain as part of the description of a decision problem. Formally, a decision problem is a tuple D = (A, E, u, Pl), where

• A = (A, S, C) is a decision situation,

• E = (U, P, V, ⊕, ⊗) is an expectation domain, • u : C → U is a utility function, and

• Pl : 2S _{→ P is a plausibility measure.}

We say that D is monotonic (resp., has a standard ⊕ identity) iff E is monotonic (resp., has a standard ⊕ identity).

2.2 (Generalized) Expected Utility

Let D = ((A, S, C), E, u, Pl) be a decision problem. Each a ∈ A induces a utility random variable ua : S → U as follows: ua(s) = u(a(s)). In the standard setting (where utilities are real-valued

and Pl is a probability measure Pr), we can identify the expected utility of act a with the expected value of ua with respect to Pr, computed in the standard way (where we use ran(f ) to denote the

range of a function f ):

EPr(ua) =

X

x∈ran(ua)

Pr(u−1_a (x)) × x.3 (2.3)

We can generalize (2.3) to an arbitrary expectation domain E = (U, P, V, ⊕, ⊗) by replacing +, ×, and Pr by ⊕, ⊗, and Pl, respectively. This gives us

EPl,E(ua) =

M

x∈ran(ua)

Pl(u−1_a (x)) ⊗ x. (2.4)

We call (2.4) the generalized EU (GEU) of act a. Clearly (2.3) is a special case of (2.4).

In the probabilistic case, if all singleton sets are measurable with respect to Pr (i.e., in the domain of Pr), then EPr(ua) = X s∈S Pr(s) × ua(s). (2.5) 3

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The plausibilistic analogue of (2.5) is not necessarily equivalent to (2.4). A decision problem ((A, S, C), E, u, Pl) is additive iff, for all c ∈ C and nonempty X, Y ⊆ S such that X ∩ Y = ∅,

Pl(X ∪ Y ) ⊗ u(c) = (Pl(X) ⊗ u(c)) ⊕ (Pl(Y ) ⊗ u(c)).

Note that the notion of additivity we defined is a joint property of several components of a de-cision problem (i.e., ⊕, ⊗, u, and Pl) instead of being a property of Pl alone. Additivity is exactly the requirement needed to make the analogue of (2.5) equivalent to (2.4). While decision problems involving probability are additive, those involving other representations of uncertainty (e.g., Dempster-Shafer belief functions or, more generally, Choquet capacities) are, in general, not additive.

Example 2.3 For a decision problem (A, E, u, Pr), where E is the standard expectation domain and u is a real-valued utility function, GEU agrees with EU.

Example 2.4 Consider the decision problem(A, E2, u, (Pr1, Pr2)), where E2 is the expectation

domain described in Example 2.2 and u is a real-valued utility function. The pair (Pr1, Pr2) of

probability measures can be viewed as a single plausibility measure. If A = (A, S, C), then the plausibility of X ⊆ S is a pair (Pr1(X), Pr2(X)). It is easy to check that

E_(Pr₁_,Pr₂_),E₂(ua) = (EPr1(ua), EPr2(ua)).

Moreover, E_(Pr₁_,Pr₂_),E₂(ua) -V E(Pr1,Pr2),E2(ua0) iff EPri(ua) ≤ EPri(ua0) for i = 1, 2.

On the other hand, if we consider E₂0, we still have E(Pr1,Pr2),E02(ua) = (EPr1(ua), EPr2(ua)), but

now E_(Pr₁_,Pr₂_),E0

2(ua)

-0

V E(Pr1,Pr2),E20(ua0) iff min(EPr1(ua), EPr2(ua)) ≤ min(EPr1(ua0), EPr2(ua0)).

We can think of the plausibility measure (Pr1, Pr2) as describing a situation where the DM is

unsure which of Pr1 and Pr2 is the “right” probability measure. In E2, act a is considered at least

as good as a0 if it is at least as good no matter which of Pr1 and Pr2 describes the actual situation.

In E₂0, a is at least as good as a0 if the worst expected outcome of a (with respect to each of Pr1

and Pr2) is at least as good as the worst expected outcome of a0.

Of course, there is nothing special about having two probability measures: we could easily allow for an arbitrary set of probability measures.

Another example of the use of GEU can be found in the proof of Theorem 3.1. Although the construction does not correspond to any standard decision problem in the literature, it does show the flexibility of the approach.

3 Representing Arbitrary Preference Relations

In this section, we show that every preference relation on acts has a GEU representation. GEU, like all decision rules, is formally a function from decision problems to preference relations on acts. Thus a GEU representation of a preference relation -A on the acts in A = (A, . . .) is a decision problem

D = (A, E, u, Pl), where E = (U, P, V, ⊕, ⊗), such that a1-Aa2 iff EPl,E(ua1) -V EPl,E(ua2).

Theorem 3.1 Every preference relation -A has a GEU representation.

Proof: Fix some A = (A, S, C) and -A. We want to construct a decision problem D = (A, E, u, Pl)

such that GEU(D) = -A.

The idea is to let each consequence be its own utility and each set be its own plausibility, and define ⊗ and ⊕ such that each act is its own expected utility. For each c ∈ C, let ac denote the

constant act with consequence c; that is, ac(s) = c for all s ∈ S. Let E = (U, P, V, ⊕, ⊗) be defined

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1. U = (C, -C), where c -C d iff c = d or ac, ad∈ A and ac-Aad. (Note that Savage assumes

that A contains all simple acts; in particular, A contains all constant acts. We do not assume that here.)

2. P = (2S, ⊆).

3. V = (2S×C, -V), where x -V y iff x = y or x, y ∈ A and x -Ay. (Note that set-theoretically

a function is a set of ordered pairs, so A ⊆ 2S×C_.)

4. x ⊕ y = x ∪ y for x, y ∈ V .

5. X ⊗ c = X × {c} for X ∈ 2S (= P ) and c ∈ C (= U ).

We can identify c ∈ C with S × {c} in V ; with this identification, (U, -U) is a substructure of

(V, -V) and > ⊗ c = c for all c ∈ U (= C), as required. Furthermore, ⊕ is clearly associative and

commutative, so E is indeed an expectation domain. Let D = (A, E, u, Pl), where u(c) = c and Pl(X) = X. Note that EPl,E(ua) = M x∈ran(ua) Pl(u−1_a (x)) ⊗ x = M c∈ran(a)Pl(a −1 (c)) ⊗ c = {(s, c) | a(s) = c} = a.

That is, each act is its own expected utility; by the definition of -V, it is clear that a -A b iff

EPl,E(ua) -V EPl,E(ub). Thus GEU(D) = -A, as desired.

Note that, unlike most representation theorems, there is no uniqueness condition in Theorem 3.1. One reason for the lack of uniqueness in Theorem 3.1 is because we place no restriction whatsoever on -A. The other reason for the lack of uniqueness is that we consider arbitrary expectation domains

instead of restricting ourselves to E. Note that, even if -A satisfies all of Savage’s postulates,

although there is a unique GEU representation of -A using the standard expectation domain E

and probability measures (this is essentially Savage’s result), there is no unique GEU representation if we allow arbitrary expectation domains. In particular, the representation constructed in the proof of Theorem 3.1 is certainly distinct from the one Savage [1954] constructs. Also, it is easy to check that, even if we restrict ourselves to E and probability measures, if -A does not satisfy all of

Savage’s postulates, there could be more than one GEU representation of -A in general.

While there is no unique GEU representation, the GEU representation we construct in the proof of Theorem 3.1 is canonical in the following sense. Fix a decision situation A = (A, S, C) and a preference relation -A. Suppose D = (A, E, u, Pl) is the decision problem constructed in the proof

of Theorem 3.1 and let D0 = (A, E0, u0, Pl0) be an arbitrary GEU representation of -A, where

E0 = (U0, P0, V0,⊕,b ⊗). It is easy to check thatb

• for all X, Y ⊆ S, Pl(X) _P Pl(Y ) implies Pl0(X) P0 Pl0(Y ),

• for all c, d ∈ C, u(c) -U u(d) implies u0(c) -U0 u0(d), and

• for all X1, . . . , Xn, Y1, . . . , Ym ⊆ S, for all c1, . . . , cn, d1, . . . , dm∈ C,

Pl(X1) ⊗ u(c1) ⊕ · · · ⊕ Pl(Xn) ⊗ u(cn) -V Pl(Y1) ⊗ u(d1) ⊕ · · · ⊕ Pl(Ym) ⊗ u(dm)

implies

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Thus, the representation we construct is minimal, in the sense that we relate only what has to be related to satisfy the definition of representation.

Note that the representation constructed in Theorem 3.1 is additive, since if X ∩ Y = ∅, then (X ∪ Y ) × {c} = (X × {c}) ∪ (Y × {c}), and has a standard ⊕ identity, namely ∅; the expectation domain E constructed in the proof of Theorem 3.1 is not (necessarily) monotonic, since we certainly could have two acts a and b such that a -Ab, so EPl,E(ua) -V EPl,E(ub), but there is some x ∈ V

such that EPl,E(ua) ⊕ x 6-V EPl,E(ub) ⊕ x. In fact, our construction has the property that two

distinct expressions are unrelated unless they are both expected utility values. As the following corollary shows, it is not hard to modify the proof by extending -V so as to make E monotonic.

Corollary 3.2 Every preference relation has a monotonic additive GEU representation with a standard ⊕ identity.

Proof: See the appendix.

Corollary 3.2 shows that requirements like monotonicity, additivity, and having a standard ⊕ identity do not restrict the kind of preference relation that GEU can represent.4 (With a little more effort, we can even build a valuation domain that is a field, and require that the plausibility domain is a substructure of the valuation domain.) But this means that these requirements do not by themselves prevent GEU from producing “strange” preference relations when it is applied as a decision rule. In the next section, we consider constraints on expectations domains that do force the preference relation produced by GEU to be arguably more reasonable.

Theorem 3.1 holds in large part because of the flexibility we have. Given a decision situation (A, S, C) and a preference relation -A on A, we are able to construct an expectation domain and

a relation -V that is customized to capture the relation -Aon A. We do not need quite this much

flexibility. We can strengthen Theorem 3.1 to show that for every decision situation A = (A, S, C), there exists an expectation domain EA such that for all preference relations -A on A, there exists

a utility function u and plausibility measure Pl such that GEU((A, E, u, Pl) = -A. That is, given

A, we can fix the expectation domain once and for all, rather than taking a different expectation structure (more precisely, a different order -V on the valuation domain) for each preference relation

-A.5 Indeed, we can even fix the plausibility measure once and for all as well.

Theorem 3.3 Given a decision situation A = (A, S, C), there exists a monotonic, additive expec-tation domain E with a standard ⊕ identity and a plausibility measure Pl on S such that, for every preference relation -A on A, there exists a utility function u-A on C and that -A = GEU(D),

where D = (A, E, u_-_A, Pl).

Proof: Again, the argument proceeds by modifying the construction in Theorem 3.1. We leave details to the appendix.

Theorems 3.1 and 3.3 depend (in part) on two features of our setup. The first is that, following Savage [1954], we took acts to be functions from states to consequences. This is not an entirely trivial assumption. As has often been observed before, one problem is that it may be unreasonable to identify consequences across different states of nature. The way a DM feels about receiving $10 may well depend on the state. But even if consequences can be identified across states, a DM might view different acts as producing the same consequences, depending on how the consequences

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However, they do affect what kind of decision rules GEU can represent.

5_{Using standard constructions from set theory, we can go even further by constructing a single expectation domain}

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are modeled. For example, suppose that Alice has a red umbrella and a blue umbrella (both in good condition). If the set of consequences is {“getting wet”, “staying dry”}, then carrying the red umbrella will produce the same consequences as carrying the blue umbrella. We can deal with the latter problem by having a consequence function c : A × S → C that takes an act a and a state s and gives the consequence of a in s. Of course, in this setting, two distinct acts a1 and a2 could

induce the same function from states to consequences; that is, we might have c(a1, s) = c(a2, s) for

all s ∈ S. It is easy to see that if a1 and a2 induce the same function from states to consequences,

then no matter what expectation domain, utility function, and plausibility measure we use, a1 and

a2 will have the same expected utility. Thus, if -A does not treat a1 and a2 the same way, then

-A has no GEU representation. To get an analogue of Theorem 3.1 in this case, we must require

that two acts that induce the same function are treated the same way by -A.

A second reason that we do not need any constraints on -Ais that we have placed no constraints

on -V, and relatively few constraints on ⊕, ⊗, u, and Pl. If, for example, we required -V to be

transitive, then we would also have to require that -A be transitive. The lack of constraints on

⊕, ⊗, u, and Pl is important because it gives us enough freedom to ensure that distinct acts have different expected utility. In the next section, we investigate what happens when we add more constraints.

4 Representing Savage’s Postulates

Theorem 3.1 shows that GEU can represent any preference relation. We are typically interested in representing preference relations that satisfy certain constraints, or postulates. The goal of this section is to examine the effect of such constraints on the components that make up GEU. For definiteness, we focus on Savage’s postulates.

A set Pe of axioms about (i.e., constraints on) decision problems represents a set of postulates

Pr about decision situation and preference relation pairs with respect to a collection of decision

problems Π iff for all D ∈ Π,

D = (A, E, u, Pl) satisfies P_e iff (A, GEU(D)) satisfies Pr.

Theorem 3.1 can be viewed as saying that the empty set of axioms represents the empty set of postulates with respect to the collection of all decision problems. Note that if Pe represents Pr

with respect to Π0 and Π1 ⊆ Π0, then Pe represents Pr with respect to Π1 as well.

Before we present Savage’s postulates, we first introduce some notation that will make the exposition more succinct. Suppose that f : X → Y , g : X → Y , and Z ⊆ X. Let hf, Z, gi denote the function h such that h(x) = f (x) for all x ∈ Z and h(x) = g(x) for all x ∈ Z. For example, if X = Y = R and Z = {x | x < 0}, then h−x, Z, xi is the absolute value function. In the intended application, the functions in question will be acts. So a = ha1, X, a2i is the act such that

a(s) = a1(s) for all s ∈ X and a(s) = a2(s) for all s ∈ X. For brevity, we identify the consequence

c ∈ C with the constant act ac such that ac(s) = c for all s ∈ S. So for c1, c2 ∈ C, hc1, X, c2i is

the act with the property that a(s) = c1 for all s ∈ X and a(s) = c2 for all s ∈ X. Recall that

X1, . . . , Xn is a partition of Y iff SiXi = Y and for all 1 ≤ i, j ≤ n such that i 6= j, Xi 6= ∅ and

Xi∩ Xj = ∅.

Fix some decision situation (A, S, C). Readers familiar with [Savage 1954] will recall that Savage assumes that A consists of all possible functions from S to C, since the DM can be questioned about any pair of functions. This is a rather strong assumption. It means that the DM is required to have preferences on a rather large set of acts, many of which are not in her power to perform (and, indeed, many of which might be impossible to realize). Savage needs this assumption for

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his theorem. We do not need it for our results, although making this assumption simplifies the statement of the relevant axioms. As we have throughout this paper, in this section, we continue to allow A to be any nonempty subset of the set of all simple acts. The reader might wonder why we do not simply allow A to be the set of all simple acts, since we do not require -A to be total.

The point is that having A consist of all simple acts conceptually requires that the DM explicitly decide, for each pair of acts, whether they are related, and if so, how; if A is a subset of the set of all acts, then the DM does not have to express a preference between acts not in A.

It turns out that the statement of a number of our results is simpler if A consists of all simple acts. To facilitate the comparison of our results with the standard results from the literature, where it is typically assumed that A consists of all simple acts, we use brackets (i.e., “

_[

” and “

_]

”) to delimit parts of the postulates that pertain to the general case in which A is an arbitrary nonempty subset of the set of all simple acts. So there are two versions of the postulates, one for the general case, which we refer to as the general version, and one for the special case (i.e., the case in which A is the set of all simple acts), which we refer to as the special version. The general version includes the bracketed statements while the special version does not. Typically, the statements inside the brackets turn unconditional assertions of the special version into implications whose antecedent says that the acts in question are in fact members of A. We recommend that the reader ignore the material inside the brackets on a first pass. Savage’s first six postulates are given in Figure 1. It is easy to check that all the bracketed statements are trivially true if A is the set of all simple acts.

As is standard in the literature, we use “a1 ≺A a2” to abbreviate “a1 -A a2 and a2 6-A a1”,

and we use “a1∼Aa2” to abbreviate “a1-Aa2 and a2 -Aa1”. (Note that in general ≺Aand ∼A

are not necessarily transitive, since -A is not necessarily transitive.)

We now give a brief overview of the intuition behind the postulates and how Savage uses them. P1 is the standard necessary condition for representation by EU (and many of its generalizations), since R is a linear order; it basically says that -Ais a total preorder. Savage defines for each subset

X ⊆ S a conditional preference relation on acts as follows: a1 -XA a2 iff

[

there exists a ∈ A such

that hai, X, ai ∈ A for i ∈ {1, 2} and

]

for all a ∈ A,

_[

if hai, X, ai ∈ A for i ∈ {1, 2}, then

]

ha1, X, ai -Aha2, X, ai.

(As in the statements of the postulates, we use brackets to delimit parts that are needed for the general version.) Intuitively, a1 -XA a2 if when X occurs the DM would find a2 at least as good

as a1. Note that -A = -SA, so P1 guarantees that -SA is a total preorder. However, -XA is not

necessarily a total preorder for all X, even if P1 holds—for this, we need P2.

P2 says that the way two acts are related depends only on where they differ; the part on which they agree can be ignored. Note that it follows from P2 that either

• for all a ∈ A,

_[

]

ha1, X, ai -Aha2, X, ai or

• for all a ∈ A, ha1, X, ai 6-Aha2, X, ai.

Thus, in the presence of P2, a1 -X_A a2 iff

_[

]

ha1, X, ai ≺Aha2, X, ai or

_[

]

ha1, X, ai ∼Aha2, X, ai.

Using -XA, Savage defines what it means for a set to be null: a set X is null iff for all a1, a2 ∈ A,

a1 -XA a2

[

iff a2 -XA a1

]

. It is easy to check that, if -XA is a total preorder, then the general

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P1. For all a1, a2, a3∈ A,

(a) a1-Aa2 or a2-Aa1, and

(b) if a1-Aa2 and a2 -Aa3, then a1 -Aa3.

P2. For all X ⊆ S, a1, a2, b1, b2 ∈ A,

[

if hai, X, bji ∈ A for i, j ∈ {1, 2}, then

]

X occurs. P3 basically says that if X is not null, then c1 -A c2 iff c1 -X_A c2. That is, whenever

the DM has some nontrivial preference, the preferences over consequences remain the same as the unconditional ones. Savage defines a relation -S on events as follows: X -SY iff

for all c, d ∈ C, if

_[

c, d ∈ A,

_]

d ≺Ac,

[

and hc, X, di, hc, Y, di ∈ A,

]

then hc, X, di -Ahc, Y, di.

The intuition is that, given two consequences c and d such that d ≺Ac, the DM prefers a binary

act that is more likely to yield c than d, according to her beliefs. This is very much in the spirit of arguments of de Finetti [1931]. P4, in the presence of P1–P3 and the assumption that A is the set of all simple acts, basically ensures that -S is a total preorder. P5 says that S is not null.

That is, the DM has some nontrivial (unconditional) preference. P1–P5 by themselves do not allow the construction of a unique EU representation (even if we assume that A is the set of all simple acts). However, with the assumption that A is the set of all simple acts, P1–P5 ensure that -S is

a qualitative probability. In order to obtain a unique EU representation we need P6, which says roughly that for all pairs of acts a, b ∈ A and consequences c ∈ C, if a ≺Ab then we can partition

S into events such that the DM does not care if c were to happen in any element of the partition. Savage also has a seventh postulate, but it is relevant only for general (nonsimple) acts. Since we consider only simple acts, we omit it here.

It may seem that we should consider stronger versions of some postulates in the general case. For example, we might consider a version of P2 that says that if a1, a2, b1, and b2 are simple acts

(not necessarily in A) such that hai, X, bji ∈ A for 1 ≤ i, j ≤ 2, then ha1, X, b1i -A ha2, X, b1i iff

ha1, X, b2i -Aha2, X, b2i. Fortunately, it is not hard to show that the stronger version is equivalent

to the version that we have stated here, where a1, a2, b1, and b2 are required to be in A, since

if ha, X, bi ∈ A, then in fact there exist some a0, b0 ∈ A such that ha0, X, b0i = ha, X, bi: just take a0= b0= ha, X, bi. Thus it suffices in all the postulates to quantify over A instead of over all simple acts.

Given a decision problem D = ((A, S, C), E, u, Pl) and ∅ 6= Z ⊆ S, define the GEU of act a restricted to Z as follows:

EPl,E(ua Z) =

M

x∈ua(Z)

Pl(u−1_a (x) ∩ Z) ⊗ x,

Note that EPl,E(ua S) = EPl,E(ua). Suppose that D = ((A, S, C), E, u, Pl) is additive. It is then

easy to check that, for all nonempty proper subsets X of S,

EPl,E(ua) = EPl,E(ua S) = EPl,E(ua X) ⊕ EPl,E(ua X),

and, more generally, given a partition X1, . . . , Xn of Y ⊆ S, we have that

EPl,E(ua Y ) = EPl,E(ua X1) ⊕ · · · ⊕ EPl,E(ua Xn).

Also, it is easy to check that for all nonempty proper subsets X of S, EPl,E(uha1,X,a2i) = EPl,E(ua1 X) ⊕ EPl,E(ua2 X).

Note that while these statements are true for additive decision problems, they are not true in general.

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A1. For all x, y, z ∈ E (S), (a) x -V y or y -V x, and

(b) if x -V y and y -V z, then x -V z.

A2. For all nonempty proper subsets X of S, x1, x2 ∈ E(X), y1, y2 ∈ E(X),

[

if xi⊕ yj ∈ E(S) for

i, j ∈ {1, 2}, then

_]

x1⊕ y1 -V x2⊕ y1 iff x1⊕ y2 -V x2⊕ y2.

A3. For all nonempty proper subsets X of S, if there exist x1, x2 ∈ E(X) such that

[

there exists

y0 ∈ E(X) such that xi⊕ y0 ∈ E(S) for i ∈ {1, 2}, and

]

for all y ∈ E (X),

_[

if xi⊕ y ∈ E(S) for i ∈ {1, 2}, then

]

x1⊕ y ≺V x2⊕ y,

u1 ≺V u2.

A6. For all x, y ∈ E (S), u ∈ ran(u), if x ≺V y, then for all a, b ∈ A, c ∈ C, such that EPl,E(ua) = x,

EPl,E(ub) = y, and u(c) = u, there exists a partition Z1, . . . , Zn of S, such that x can be

expressed as x1⊕ · · · ⊕ xn and y can be expressed as y1⊕ · · · ⊕ yn, where xk= EPl,E(ua Zk)

and yk= EPl,E(ub Zk) for 1 ≤ k ≤ n, and for all 1 ≤ i ≤ n,

[

if Pl(Zi) ⊗ u ⊕L_k6=ixk ∈ E(S) then

]

Pl(Zi) ⊗ u ⊕L_k6=ixk≺V y and

[

if Pl(Zi) ⊗ u ⊕Lk6=iyk∈ E(S) then

]

x ≺V Pl(Zi) ⊗ u ⊕Lk6=iyk.

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Let ED(X) = {EPl,E(ua X) | a ∈ A}. (We omit the subscript D if it is clear from context.)

Intuitively, ED(X) consists of all the expected utility values of acts in A restricted to X. To simplify

the statement of one of the axioms, let

hhu, X, vii =    u if X = S, v if X = ∅, Pl(X) ⊗ u ⊕ Pl(X) ⊗ v otherwise,

where u, v ∈ U and X ⊆ S. Note that EPl,E(uhc,X,di) = hhu(c), X, u(d)ii. The cases X = ∅ and

X = S must be treated specially, since we do not assume that ⊥ ⊗ u is the identity for ⊕. As with Savage’s postulates, we use brackets to delimit parts needed for the general version. See Figure 2 for a list of the axioms.

A1 says that the expected utility values are linearly preordered; more specifically, A1a says that all pairs are comparable and A1b says that the relation is transitive. Note that A1 does not say that the whole valuation domain is linearly preordered: that would be a sufficient but not a necessary condition for GEU(D) to satisfy P1. Since we want necessary and sufficient conditions for our representation results, some axioms apply only to expected utility values rather than to arbitrary elements of the valuation domain.

[

There is a technical assumption that we need for some parts of the general version of our result. In general, it might be the case that EPl,E(ua) ∈ E (S), but a /∈ A; this could happen if,

even though a /∈ A, there is some b ∈ A such that E_Pl,E(ua) = EPl,E(ub). (Note that ua is well

defined whether or not a ∈ A, so it makes sense to talk about EPl,E(ua) even if a /∈ A.) We say

that D is whole iff this does not happen; more precisely, D = ((A, S, C), E, u, Pl) is whole iff for all simple acts a ∈ CS, EPl,E(ua) ∈ E (S) implies a ∈ A. A decision problem whose set of acts is the

set of all simple acts is whole, but that is not a necessary condition for a decision problem to be whole. In general, a decision problem D = ((A, S, C), E, u, Pl), where E = (U, P, V, ⊕, ⊗) is whole iff, for all x ∈ V , either every act with expected utility x is in A, or no act with expected utility x is in A.

_]

To simplify the statement of the theorem, let Πall be the collection of all decision problems;

let Πadd be the collection of additive decision problems; and let Π0 be the collection of decision

problems whose set of acts is the set of all simple acts

_[

together with all decision problems that are whole

_]

. Let

• Π_1a= Π1b= Πall,

• Π4 = Π5 = Π0,

• Π₂ = Π3 = Π6= Πadd∩ Π0.

Theorem 4.1 For all i1, . . . , ik ∈ {1a, 1b, . . . , 6}, {Ai1, . . . , Aik} represents {Pi1, . . . , Pik} with

respect to Πi1∩ · · · ∩ Πik.

Proof: See the appendix.

Theorem 4.1 is a strong representation result. For example, if we are interested in capturing all of Savage’s postulates but the requirement that -A is a total preorder, and instead are willing

to allow it to be a partial preorder (a situation explored by Lehmann [1996]), we simply need to drop the axiom A1a. Although we have focused here on Savage’s postulates, it is straightforward to represent many of the other standard postulates considered in the decision theory literature in much the same way.

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5 Conclusion

We have introduced GEU, a notion of generalized EU, and shown that GEU can (a) represent all preference relations on acts and (b) be customized to capture any subset of Savage’s postulates. As we pointed out in the introduction, these results may be of particular interest to designers of software agents, who may want to deal with more general representations of tastes and beliefs than real-valued utilities and probabilities. If beliefs are represented using a plausibility measures and tastes are represented by a utility function that is not necessarily real-valued, the problem for the software designer is reduced to finding appropriate ways of combining plausibility and utility using ⊕ and ⊗, and finding an appropriate binary relation - on the resulting expressions.

The results of this paper suggest that rationality postulates can be captured by choosing ⊕, ⊗, and -, so that they satisfy certain constraints. The results of [Chu and Halpern 2003] show that we lose no generality by using GEU to represent the decision making process; essentially all decision rules can be (ordinally) represented by GEU. Thus, the framework of expectation domains together with GEU provides a useful level of abstraction in which to study the general problem of decision making and rules for decision making and a useful conceptual framework for designing decision rules for software agents.

Acknowledgments

We thank Duncan Luce, Leandro Rego, and Peter Wakker for useful comments on a draft of the paper.

A

Proofs

Corollary 3.2 Every preference relation has a monotonic additive GEU representation with a standard ⊕ identity.

Proof: Fix some A = (A, S, C) and -A. Let D be as defined in the proof of Theorem 3.1, except

that we change the definition of -V to x -V y iff x = y or there exist a, b ∈ A such that a -A b

and

1. x = EPl,E(ua) and y = EPl,E(ub), or

2. x = EPl,E(ua) ⊕ z and y = EPl,E(ub) ⊕ z for some z ∈ V .

Recall that EPl,E(ua) = a, so without case 2, we are back in the situation described in the proof

of Theorem 3.1. Note that, by construction, the only way that EPl,E(ua) ⊕ z can be an expected

utility value is if z ⊆ EPl,E(ua), since ⊕ = ∪ and proper supersets of expected utility values cannot

be expected utility values. Thus, if in case 2 both x and y are expected utility values, then we must in fact be in case 1; case 2 has an effect only when x and y are not both expected utility values. Thus, case 2 does not affect how pairs of expected utility values are related, so we still have that GEU(D) = -A.

To see that ⊕ is monotonic with respect to this definition of -V, suppose that x -V y. We

need to show that x ⊕ z -V y ⊕ z. If x = y, then x ⊕ z = y ⊕ z, so the conclusion holds. Suppose

that x 6= y. Then there exist some a, b ∈ A such that a -Ab and either case 1 or case 2 holds. It

is easy to see that in either case, x ⊕ z -V y ⊕ z by case 2. Thus D is a monotonic representation

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Theorem 3.3 Given a decision situation A = (A, S, C), there exists a monotonic, additive expec-tation domain E with a standard ⊕ identity and a plausibility measure Pl on S such that, for every preference relation -A on A, there exists a utility function u-A on C and that -A = GEU(D),

where D = (A, E, u_-_A, Pl).

Proof: Let P(A) consist of all preference relations on A. We now modify the construction in Theorem 3.1 as follows:

1. U = (C × 2P(A), -U), where (c, X) -U (d, Y ) iff X = {-A} = Y for some -A∈ P(A) and

either c = d or ac, ad∈ A and ac-Aad.

2. P = (2S, ⊆).

3. V = (2S×C × 2P(A)_,

-V), where (x, X) -V (y, Y ) iff X = {-A} = Y for some -A ∈ P(A)

and either x = y or x, y ∈ A and x -Ay.

4. (x, X) ⊕ (y, Y ) = (x ∪ y, X ∪ Y ).

5. X ⊗ (c, Y ) = (X × {c}, Y ) for X ⊆ S, c ∈ C, and Y ⊆ P(A).

The same arguments as in the proof of Theorem 3.1, this construction gives an additive expectation domain. We can modify -V as in Corollary 3.2 to make it monotonic. With a little more effort,

we can further modify it so that there is a standard ⊕ identity; we omit details here.

Let Pl(X) = X. Given a preference relation -A, define the utility function u-A by taking

u_-_A(c) = (c, {-A}). Again, the same arguments as those in Theorem 3.1 can be used to show that

-A= GEU(D), where D = (A, E, u-A, Pl).

Theorem 4.1 For all i1, . . . , ik ∈ {1a, 1b, . . . , 6}, {Ai1, . . . , Aik} represents {Pi1, . . . , Pik} with

respect to Πi1∩ · · · ∩ Πik.

Proof: We first establish the result for singleton sets. Let D = (A, E, u, Pl), where A = (A, S, C), be an arbitrary decision problem.

_[

As in the statements of the postulates and axioms, we will use brackets to delimit the parts of the proof that pertain to the conditional versions.

_]

• A1a represents P1a and with respect to Π_1a and A1b represents P1b with respect to Π1b.

We do the case of A1a represents P1a with respect to Π1a and leave the other case, which is

completely analogous to the one we do, to the reader.

Suppose that D satisfies A1a. We need to show that (A, GEU(D)) satisfies P1a. Let a1, a2, a3∈ A. Let xi = EPl,E(uai); clearly x1, x2, x3∈ E(S). Since D satisfies A1a, x1-V x2

or x2 -V x1. In other words, a1 -GEU(D) a2 or a2 -GEU(D) a1. Thus (A, GEU(D)) satisfies

P1a. Now suppose that (A, GEU(D)) satisfies P1a. We need to show that D satisfies A1a. Let x1, x2, x3 ∈ E(S). Then there exist a1, a2, a3 ∈ A such that EPl,E(uai) = xi. Since

(A, GEU(D)) satisfies P1a, a1 -GEU(D) a2 or a2 -GEU(D) a1. Thus x1 -V x2 or x2 -V x1.

So D satisfies A1a.

• A2 represents P2 with respect to Π2.

Throughout this part of the proof, we assume that D ∈ Π2; in particular, we assume that D

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Suppose that D satisfies A2. We need to show that (A, GEU(D)) satisfies P2. Suppose that X ⊆ S and a1, a2, b1, b2 ∈ A.

[

Suppose further that hai, X, bji ∈ A.

]

We need to show that

ha₁, X, b1i -GEU(D) ha2, X, b1i iff ha1, X, b2i -GEU(D)ha2, X, b2i.

If X = ∅ or X = S, then the above is trivially true. So assume that X is a nonempty proper subset of S. Let xi = EPl,E(uai X) and yj = EPl,E(ubj X). Clearly xi ∈ E(X), yj ∈ E(X),

and EPl,E(uhai,X,bji) = xi⊕ yj.

[

Furthermore, xi⊕ yj ∈ E(S), since hai, X, bji ∈ A.

]

Since D

satisfies A2, we have that

x1⊕ y1 -V x2⊕ y1 iff x1⊕ y2 -V x2⊕ y2,

so

ha1, X, b1i -GEU(D) ha2, X, b1i iff ha1, X, b2i -GEU(D)ha2, X, b2i.

Thus (A, GEU(D)) satisfies P2.

Suppose that (A, GEU(D)) satisfies P2. We need to show that D satisfies A2.

_[

We assume for this direction that D is whole.

_]

Suppose that X is a nonempty proper subset of S, x1, x2 ∈ E(X), and y1, y2 ∈ E(X).

[

Suppose further that xi⊕ yj ∈ E(S).

]

We need to show

that

x1⊕ y1 -V x2⊕ y1 iff x1⊕ y2 -V x2⊕ y2.

Note that there exist a1, a2, b1, b2∈ A such that EPl,E(uai X) = xi and EPl,E(ubi X) = yi.

Observe that EPl,E(uhai,X,bji) = xi⊕ yj.

[

Since xi⊕ yj ∈ E(S) and D is whole, it follows that

ha_i, X, bji ∈ A.

]

Since (A, GEU(D)) satisfies P2,

ha₁, X, b1i -GEU(D) ha2, X, b1i iff ha1, X, b2i -GEU(D)ha2, X, b2i,

so

x1⊕ y1 -V x2⊕ y1 iff x1⊕ y2 -V x2⊕ y2.

Thus D satisfies A2.

is additive and we will use this fact without further comment.

For this part, we will prove a slightly stronger claim that actually has a shorter proof. Note that A3 and P3 are both implications. We show that D satisfies the antecedent (consequent) of A3 iff (A, GEU(D)) satisfies the antecedent (consequent) of P3.

_[

We assume that D is whole in these arguments.

_]

This implies that D satisfies A3 iff (A, GEU(D)) satisfies P3. Note that P3 quantifies over all subsets of S while A3 quantifies over only nonempty proper subsets of S. It is easy to check that ∅ and S satisfy P3. (More precisely, (A, GEU(D)) satisfies the instance of P3 in which X is instantiated with ∅ and the instance of P3 in which X is instantiated with S.) So for the rest of this part, we restrict our attention to nonempty proper subsets of S.

We begin by showing that D satisfies the antecedent of A3 iff (A, GEU(D)) satisfies the antecedent of P3. Fix some X that is a nonempty proper subset of S. We need to show that there exist x1, x2∈ E(X) such that

1.

_[

there exists y0 ∈ E(X) such that xi⊕ y0 ∈ E(S) and

]

[

if xi⊕ y ∈ E(S),

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iff there exist a1, a2∈ A such that

2.

_[

there exists b0 ∈ A such that hai, X, b0i ∈ A and

]

for all b ∈ A,

[

if hai, X, bi ∈ A, then

]

ha1, X, bi ≺Aha2, X, bi.

To see that 1 implies 2, suppose that x1, x2∈ E(X) satisfy 1. Since x1, x2 ∈ E(X), there exist

a1, a2 ∈ A such that EPl,E(uai X) = xi. We show that a1 and a2 satisfy 2.

[

To see that

the first conjunct is true, note that by 1 there exists y0 ∈ E(X) such that xi⊕ y0 ∈ E(S);

fix some such y0. Note that there exists b0 ∈ A such that EPl,E(ub0 X) = y0; observe that

EPl,E(uhai,X,b0i) = xi⊕ y0 ∈ E(S). Since D is whole, hai, X, b0i ∈ A. For the second conjunct,

we proceed as follows.

_]

Let b ∈ A

_[

be such that hai, X, bi ∈ A

]

. We need to show that

ha1, X, bi ≺A ha2, X, bi. Let y = EPl,E(ub X). Note that y ∈ E(X) and EPl,E(uhai,X,bi) =

xi ⊕ y.

[

Furthermore, since hai, X, bi ∈ A, xi⊕ y ∈ E(S).

]

By 1, x1 ⊕ y ≺V x2 ⊕ y; thus

ha₁, X, bi ≺Aha2, X, bi.

To see that 2 implies 1, suppose that a1, a2 ∈ A satisfy 2. Let xi = EPl,E(uai X); note that

xi ∈ E(X). We show that x1and x2satisfy 1.

[

To see that the first conjunct is true, note that

by 2 there exists b0 ∈ A such that hai, X, b0i ∈ A; fix some such b0. Let y0 = EPl,E(ub0 X).

Note that xi⊕y0 = EPl,E(uhai,X,bi), so y0∈ E(X) and xi⊕y0∈ E(S). For the second conjunct,

u1 -V u2 iff

[

there exists y ∈ E (X) such

that Pl(X) ⊗ ui⊕ y ∈ E(S) and

]

hci, X, bi ∈ A and

]

for all b ∈ A,

[

if hci, X, bi ∈ A, then

]

hc1, X, bi -GEU(D)hc2, X, bi.

Suppose that 3 holds. We need to show that 4 holds. Fix some c1, c2 ∈ C

[

such that

c1, c2 ∈ A

It is easy

to check that there exists y ∈ E (X) such that Pl(X) ⊗ ui⊕ y ∈ E(S) iff there exists b ∈ A such

that hci, X, bi ∈ A; the “only if” part depends on the assumption that D is whole.

]

To see that

4 holds, fix some b ∈ A

_[

such that hci, X, bi ∈ A

]

hc_i, X, bi ∈ A and

_]

for all b ∈ A,

_[

if hci, X, bi ∈ A, then

]

hc1, X, bi -GEU(D) hc2, X, bi.

[

As

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y ∈ E (X) such that Pl(X) ⊗ ui⊕ y ∈ E(S); now the “if” part depends on the assumption that

D is whole.

_]

To see that 3 holds, fix some y ∈ E (X)

_[

such that Pl(X) ⊗ ui⊕ y ∈ E(S)

]

. We

need to show that Pl(X) ⊗ u1⊕ y -V Pl(X) ⊗ u2⊕ y. Note that there exists some b ∈ A such

that EPl,E(ub X) = y; observe that EPl,E(uhci,X,bi) = Pl(X) ⊗ ui⊕ y

[

and hci, X, bi ∈ A,

since Pl(X) ⊗ ui ⊕ y ∈ E(S) and D is whole

]

. By 4, hc1, X, bi -GEU(D) hc2, X, bi; thus

Pl(X) ⊗ u1⊕ y -V Pl(X) ⊗ u2⊕ y.

Suppose that D satisfies A4. We need to show that (A, GEU(D)) satisfies P4. Suppose that X1, X2 ⊆ S, c1, d1, c2, d2 ∈ C,

[

c1, d1, c2, d2 ∈ A, hci, Xj, dii ∈ A,

]

d1 ≺A c1, and d2 ≺A c2.

We need to show that

hc1, X1, d1i -GEU(D) hc1, X2, d1i iff hc2, X1, d2i -GEU(D)hc2, X2, d2i.

Let ui = u(ci) and vi = u(di). Note that u1, v1, u2, v2 ∈ ran(u)

[

and u1, u2, v1, v2 ∈ E(S)

]

.

Also, vi ≺V ui, since di ≺GEU(D) ci. Note that EPl,E(uhci,Xj,dii) = hhui, Xj, viii.

[

Since

hc_i, Xj, dii ∈ A, hhui, Xj, viii ∈ E(S).

]

Since D satisfies A4,

hhu₁, X1, v1ii -V hhu1, X2, v1ii iff hhu2, X1, v2ii -V hhu2, X2, v2ii,

which means that

hc1, X1, d1i -GEU(D) hc1, X2, d1i iff hc2, X1, d2i -GEU(D)hc2, X2, d2i.

Thus, (A, GEU(D)) satisfies P4.

Now suppose that (A, GEU(D)) satisfies P4. We need to show that D satisfies A4.

_[

For this direction, we assume that D is whole.

_]

Suppose that X1, X2 ⊆ S, u1, v1, u2, v2 ∈ ran(u),

[

u1, v1, u2, v2∈ E(S), hhui, Xj, viii ∈ E(S),

]

v1≺V u1, and v2 ≺V u2. We need to show that

Let c1, d1, c2, d2 ∈ C be such that

[

c1, d1, c2, d2 ∈ A,

]

u(ci) = ui and u(di) = vi. Then we

see that di ≺GEU(D) ci, since vi ≺V ui. Note that hhui, Xj, viii = EPl,E(uhci,Xj,dii).

[

Since

hhu_i, Xj, viii ∈ E(S) and D is whole, hci, Xj, dii ∈ A.

]

Since (A, GEU(D)) satisfies P4,

hc₁, X1, d1i -GEU(D) hc1, X2, d1i iff hc2, X1, d2i -GEU(D)hc2, X2, d2i,

which implies that

So D satisfies A4.

• A5 represents P5 with respect to Π₅.

D satisfies A5 iff there exist u₁, u2 ∈ ran(u) such that

[

u1, u2∈ E(S) and

]

u1≺V u2 iff there

exist some c1, c2∈ C such that

[

c1, c2 ∈ A and

]

u(c1) ≺V u(c2)

[

(for the “only if” direction,

we use the assumption that D is whole)

_]

iff there exist some c1, c2 ∈ C such that

[

c1, c2∈ A

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• A6 represents P6 with respect to Π₆.

is additive and we will use this fact without further comment.

Suppose that D satisfies A6. We need to show that (A, GEU(D)) satisfies P6. Let a, b ∈ A and c ∈ C. Suppose that a ≺GEU(D) b. We need to show that there exists a partition Z1, . . . , Zn

of S, such that for all Zi,

1.

_[

if hc, Zi, ai ∈ A then

]

hc, Zi, ai ≺GEU(D) b and

2.

_[

if hc, Zi, bi ∈ A then

]

a ≺GEU(D) hc, Zi, bi.

Let x = EPl,E(ua), y = EPl,E(ub), and u = u(c). Then x, y ∈ E (S), u ∈ ran(u), x ≺V y,

EPl,E(ua) = x, EPl,E(ub) = y, and u(c) = u, so (by A6) there exists a partition Z1, . . . , Zn

of S, such that x can be expressed as x1⊕ · · · ⊕ xn and y can be expressed as y1⊕ · · · ⊕ yn,

where xk= EPl,E(ua Zk) and yk= EPl,E(ub Zk) for 1 ≤ k ≤ n, and for all 1 ≤ i ≤ n,

3.

_[

if Pl(Zi) ⊗ u ⊕L_k6=ixk ∈ E(S) then

]

Pl(Zi) ⊗ u ⊕L_k6=ixk≺V y, and

4.

_[

if Pl(Zi) ⊗ u ⊕Lk6=iyk∈ E(S) then

]

x ≺V Pl(Zi) ⊗ u ⊕Lk6=iyk.

To see that 1 holds, note that Pl(Zi) ⊗ u ⊕L_k6=ixk = EPl,E(uhc,Zi,ai).

[

Suppose that

hc, Zi, ai ∈ A; then EPl,E(uhc,Zi,ai) ∈ E (S).

]

By 3, Pl(Zi) ⊗ u ⊕

L

k6=ixk ≺V y. Thus

hc, Z_i, ai ≺GEU(D) b as desired. The argument that 2 holds is completely analogous (we use 4

instead of 3 to establish that 2 holds), and we leave it to the reader.

Now suppose that (A, GEU(D)) satisfies P6. We need to show that D satisfies A6.

_[

For this direction we assume that D is whole.

_]

Let x, y ∈ E (S) and u ∈ ran(u). Suppose that x ≺V y.

Let a, b ∈ A and c ∈ C be such that EPl,E(ua) = x, EPl,E(ub) = y, and u(c) = u. We need to

show that there exists a partition Z1, . . . , Znof S such that x can be expressed as x1⊕ · · · ⊕ xn

and y can be expressed as y1⊕ · · · ⊕ yn, where xk = EPl,E(ua Zk) and yk = EPl,E(ub Zk)

for 1 ≤ k ≤ n, and for all 1 ≤ i ≤ n, 3 and 4 hold.

Since x ≺V y, a ≺GEU(D) b, so (by P6) there exists a partition Z1, . . . , Zn of S such that for

all Zi, 1 and 2 hold. To see that 4 holds, note that EPl,E(uhc,Zi,bi) = Pl(Zi) ⊗ u ⊕

L

k6=iyk.

[

Suppose that EPl,E(uhc,Zi,bi) ∈ E (S); then hc, Zi, bi ∈ A, since D is whole.

]

By 2, a ≺A

hc, Zi, bi. Thus x ≺V Pl(Zi) ⊗ u ⊕Lk6=iyk. The argument that 3 holds is completely

analogous (we use 1 instead of 2 to establish that 3 holds), and we leave that to the reader. So far we have shown that Ai represents Pi with respect to Πi, for i ∈ {1a, 1b, . . . , 6}. Let

i1, . . . , ik ∈ {1a, 1b, . . . , 6}. Suppose that D ∈ Πi1 ∩ · · · ∩ Πik and that D satisfies {Ai1, . . . , Aik}.

Since D ∈ Πij and D satisfies Aij, it follows that (A, GEU(D)) satisfies Pij. Thus (A, GEU(D))

satisfies {Pi1, . . . , Pik}. Conversely, if D ∈ Πi1∩ · · · ∩ Πik and (A, GEU(D)) satisfies {Pi1, . . . , Pik},

then D satisfies {Ai1, . . . , Aik}. Thus {Ai1, . . . , Aik} represents {Pi1, . . . , Pik} with respect to

Πi1∩ · · · ∩ Πik.

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